Thirdly, in [1]}, for log-concave functions, the author proved that \(\nabla f\)  and \(\nabla Sf\)  are equal to the same constant in the line parallelling to \(e_n\)  on the boundaries of subgraphs of \(f\)  and \(Sf\) . For general Sobolev functions, without the convexity, we can not make use of Lemma 1.5.14 and Theorem 1.5.15 in [2]} with respect to convex functions to prove the similar conclusion. In this paper, using the more subtle method, we prove that Sobolev functions have the similar properties as log-concave functions (see Lemma REF ).
